By default your contacts are sync'd to Google. You should be able to find your contacts by logging into Google and looking at contacts under gmail.
Also when you get your replacement Android, just setting up the same account that you did for your broken android will bring all your contact to the new android.
Of course if you, for some reason, disabled the automatic sync, this method wouldn't work.
But again, without any changes or having performed special configurations on your part. This will work.
Update information for consideration:
For some (particularly me) my data is substantially more important than the hardware. A resource to consider in gaining access to your android is to actually repair it. It's more expensive than some of the suggestions and links provided by Dan Hulme, Izzy, AI E, Chahk, and andeldererathis. But it's very likely to get you your important data.
First, the cracked screen won't stop your access to your device. However, a damaged touch screen will prevent you from being able to access it. To the surest way to resolve the issue is to invest in repairing the touch screen.
You may find the replacement part on Ebay or Amazon. This is an idea of what would be involved:
Youtube: How to disassemble karbonn a15 completely:
Again, this is a hardware resolution to recovering your data. Working with the tiny parts isn't something that most of us can handle. So you'll have to search for a service center to do it for you.
While rather timely and expensive, it's not as costly as trying to hire clean room services to recover data from a hard drive.